cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A307038 Unitary imperfect numbers: numbers n that equal to their unitary analog of the alternating sum of divisors, A307037(n).

Original entry on oeis.org

1, 20, 272, 65792, 2901600, 4596800, 29016000, 4295032832, 5789534400, 49085337600, 585248256000, 960935040000
Offset: 1

Views

Author

Amiram Eldar, Mar 21 2019

Keywords

Comments

Includes all the numbers of the form F(k)*(F(k)-1) with k > 0, where F(k) = 2^(2^k) + 1, the k-th Fermat number, is prime.
a(9) > 5*10^9.
Also in the sequence are 49085337600, 585248256000, 960935040000 and 46127952000000.
a(13) > 2*10^12. - Giovanni Resta, Mar 21 2019

Crossrefs

Cf. A000215, A127724 (k-imperfect), A127725 (2-imperfect), A127726 (3-imperfect), A307037.

Programs

  • Mathematica
    f[p_,e_] := p^e + (-1)^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[10^6], a[#] == # &]

Extensions

a(9)-a(12) from Giovanni Resta, Mar 21 2019

A307051 Lesser of amicable numbers pair m < n such that A307037(m) = n and A307037(n) = m.

Original entry on oeis.org

4, 16, 68, 256, 1028, 4112, 7200, 65536, 262148, 921600, 1036800, 1048592, 16777472, 88218000, 371404800, 418535784576
Offset: 1

Views

Author

Amiram Eldar, Mar 21 2019

Keywords

Comments

The larger counterparts are in A307052.
Includes all the pairs of the form {F(k)-1, F(k)} with 0 < k, where F(k) = 2^(2^k) + 1, the k-th Fermat number, is prime.
Includes all the pairs of the form {(F(k)-1)*F(m), F(k)*(F(m)-1)} with 0 < k < m, where both F(k) and F(m) are primes.
a(17) > 4*10^12. - Giovanni Resta, Mar 24 2019

Crossrefs

Cf. A000215, A307037, A307038, A307052.

Programs

  • Mathematica
    f[p_, e_] := p^e + (-1)^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; s={}; Do[m = a[n]; If[m > n && a[m] == n, AppendTo[s, n]], {n, 1, 10^7}]; s

Extensions

a(16) from Giovanni Resta, Mar 24 2019

A307052 Larger of amicable numbers pair m < n such that A307037(m) = n and A307037(n) = m.

Original entry on oeis.org

5, 17, 80, 257, 1280, 4352, 8060, 65537, 327680, 1065220, 1089452, 1114112, 16842752, 100340800, 383479200, 466344000000
Offset: 1

Views

Author

Amiram Eldar, Mar 21 2019

Keywords

Comments

The terms are ordered according to their lesser counterparts (A307051).
a(17) > 4*10^12. - Giovanni Resta, Mar 24 2019

Crossrefs

Cf. A307037, A307038, A307051.

Programs

  • Mathematica
    f[p_, e_] := p^e + (-1)^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; s={}; Do[m = a[n]; If[m > n && a[m] == n, AppendTo[s, m]], {n, 1, 10^7}]; s

Extensions

a(16) from Giovanni Resta, Mar 24 2019

A333408 Numbers k such that s(k) = s(k+1), where s(k) is the unitary analog of the alternating sum-of-divisors function (A307037).

Original entry on oeis.org

1, 11, 71, 155, 194, 204, 1119, 1420, 1628, 3705, 5186, 14363, 19788, 20467, 25180, 25545, 25843, 28251, 30804, 42811, 60204, 63180, 71791, 73260, 83600, 87219, 87308, 91539, 97432, 99208, 100456, 100471, 119315, 122616, 132308, 135340, 135864, 164691, 166624
Offset: 1

Views

Author

Amiram Eldar, Mar 19 2020

Keywords

Examples

			1 is a term since s(1) = s(2) = 1;
11 is a term since s(11) = s(12) = 10;

Links

Crossrefs

The unitary version of A206368.
Cf. A206369, A307037.

Programs

  • Mathematica
    f[p_, e_] := p^e + (-1)^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; seq = {}; s1 = 1;  Do[s2 = s[n]; If[s1 == s2, AppendTo[seq, n - 1]]; s1 = s2, {n, 2, 10^5}]; seq
Showing 1-4 of 4 results.

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